A 'distance' measure that involves 3 quantum states

The following question was asked by my friend Elie Wolfe.

Given two quantum (or even classical) states $\rho, \sigma$, there are various measures that say how 'far' these two quantum states are, such as fidelity, trace distance, relative entropy, Renyi relative entropies etc. While most of these measures may not be a 'distance' in strict mathematical sense (for example, fidelity does not satisfy a triangle inequality), they all share a very nice property that they are monotonic under the action of a quantum map. More precisely, let $D(\rho,\sigma)$ be any of above measures (in case of fidelity, let $D(\rho,\sigma) = -\log (F(\rho,\sigma))$) and let $\mathcal{E}$ be a trace preserving quantum map. Then it holds that

$D(\mathcal{E}(\rho),\mathcal{E}(\sigma)) <= D(\rho,\sigma).$

The question is if there is any measure $D(\rho,\sigma,\tau)$ that is

1) defined for three quantum states $\rho,\sigma,\tau$,

2) is not merely a function of measures for two quantum states and

3) is monotonic under a quantum map: $D(\rho,\sigma,\tau) >= D(\mathcal{E}(\rho),\mathcal{E}(\sigma),\mathcal{E}(\tau)).$

An intuition behind this question (that my friend gave) is the following. Consider two unit vectors in Hilbert space. Trace distance between these vectors measures the `euclidean distance' between them. Then maybe the measure for three vectors corresponds to the 'area' enclosed by tips of the vector.

• To start with, for qubits you could use the area spanned in the Bloch sphere. – Norbert Schuch May 18 '16 at 7:51
• Just some thoughts ... You could try to follow the theory of volumes in normed spaces (closely related to convex sets). But you have to be careful if there is no a triangle inequality. If you have a proper distance on the space where you have embedded your state space, then there are infinitely many ways of constructing notions of volume. One example would be the Hausdorff measure. These notions collapse to a canonical one if your distance originates from an inner product as the trace. Also these volumes share the properties that maps which do not increase distance does not increase volume. – Markus Heinrich May 18 '16 at 8:42
• Is there some operational intuition behind this? For two states, these measures typically have some interpretation, like how well we can distinguish between the states or how likely we might mistake one for the other. Then, the LOCC monotonicity is an obvious property. Is there any similar intuition for asking for a three-state monotone? With such an intuition, it might be easier to come up with such a monotone. – Norbert Schuch May 18 '16 at 16:01
• @NorbertSchuch Do you know if the area spanned in Bloch sphere would satisfy monotonicity? The motivation for this question comes from study of quantum nonlocality, in particular the recent paper : arxiv.org/abs/1507.00213 , where fidelity plays the key role. We would like to see if a measure for three quantum states gives better dimension lower bounds. Apart from this, we do not have much intuition for this question. – anurag anshu May 21 '16 at 3:42
• A good introduction is "Volumes on Normed and Finsler Spaces" by Alvarez Paiva & Thompson: library.msri.org/books/Book50/files/02AT.pdf; It requires some differential geometric background, though. The main idea is found in the first paragraphs of section 3. It could look like these ideas rely heavily on the notion of a vector space, but convex geometry and the geometry of normed spaces are in many ways related. But I don't know if this helps you, the answer to your question could be that there are many. Maybe if you demand additional properties. Let me know if I shall elaborate this. – Markus Heinrich May 23 '16 at 7:39